It has long been known that the methods of holography can be used to create records of digital data. For example, data are initially provided in the form of a two-dimensional array of elements such as spots or rectangular pixels. Each of the individual data elements can assume a binary value of 1 or 0 or represent several bits by encoding the light intensity (gray scale) transmitted through the optical system. For example, a totally opaque element may represent 0, whereas a totally transparent element may represent 1. An array of this kind has been referred to as a data mask.
The technique of Fourier transform holography relies on the physical principle that when an object is placed in the front focal plane of a converging lens, the optical field at the back focal plane corresponds to the Fourier transform of that object. (More generally, shifting the object out of the front focal plane of the lens will simply add phase terms to the Fourier transform observed at or near the back focal plane.).
The Fourier transform is a representation of the spatial characteristics of the object. Like an optical image, the Fourier transform has an amplitude that varies meaningfully from place to place. However, the amplitude at a given location in the Fourier transform does not correspond directly to, e.g., the luminance of the object at a given point (as would be the case in an image). Instead, each small region of the Fourier transform receives contributions from essentially every point on the object. As a result of the manner in which these contributions are combined, the amplitude in a given small region expresses the relative contribution that a given spatial frequency makes to the overall pattern represented by the object. Each place within the Fourier transform relates to a corresponding spatial frequency. In this sense, a record of the Fourier transform provides a spatial frequency spectrum of the object.
The Fourier transform may be recorded by placing a suitable recording medium in the back focal plane of the transforming lens. The earliest such media were photographic plates. In addition to photographic media, which are still in use, currently available media include photopolymers, as well as photochromic, photorefractive, and thermoplastic media.
The recording takes place by forming an interference pattern that impinges on the recording medium. Two light beams, referred to as the object beam and the reference beam, are used to form this interference pattern. In order to interfere, these beams must be at least partially coherent, that is, they must be at least partially correlated in phase. In many cases, these beams are generated by passing a single laser beam through a beam splitter.
An illustrative recording setup using a transmissive data mask is shown in FIG. 1. Object beam 05 is created by modulating a plane wave by data mask 10, which is, e.g., a spatial light modulator (SLM). Modulation may be transmissive, as shown, or alternatively, it may be reflective. The object beam then passes through transforming lenses 15, 20, and 25, and impinges on storage medium 30. In a typical arrangement, the lenses are spaced in a standard 4F configuration. (In such a configuration, the spacing between adjacent lenses is equal to the sum of their respective focal lengths. The spacing between a lens and an adjacent element such as data mask 10 is one focal length of that lens.) Reference beam 35 does not pass through the data mask or the system of transforming lenses, but instead is combined directly with object beam 05 on storage medium 30 to form the interference pattern that is recorded as a hologram. The object and reference beams overlap in region 40 of medium 30.
An image of the original object is reconstructed by impinging on medium 30 an excitation beam having the same angle of incidence, wavelength, or wavefront (or combination of these properties) as the reference beam used to create the hologram. Diffraction of the excitation beam by the hologram gives rise to a further, reconstructed output beam 45 that is Fourier transformed by the system of lenses 50, 55, 60 produce the image. For automatic reading of data, the image is usefully projected onto an array of sensors 65. Such an array is readily provided as, for example, a CCD array or a CMOS optical sensor array.
One practical difficulty posed by photographic emulsions and other holographic media is that none of these exhibit a perfectly linear dynamic range. That is, the optical density of the exposed medium will be proportional to the exposure for only a limited range of exposures. In addition, diffraction efficiency even in a perfect material varies with a figure of merit referred to as the modulation depth. The modulation depth at a given location within the recording medium is the intensity ratio of the object beam to the reference beam at that location.
Practitioners have observed that when the Fourier transform of an object is recorded holographically, as described here, the exposure in significant parts of the hologram that are displaced from the optical axis often tends to be much weaker in intensity than parts lying at or near the optical axis.
This occurs because in the Fourier transform plane, a significant fraction of the total illumination tends to be concentrated in a relatively small spot about the optical axis. This spot corresponds to those few spatial frequencies (generally zero and low-valued frequencies) that are highly represented in any data mask, including data masks that are inherently random in amplitude. We refer to this spot as the "dc spot", in analogy to direct electrical current (dc), which has only a zero frequency component.
If the reference beam is adjusted to match the high intensity of the dc spot, the higher frequencies will have much less diffraction efficiency relative to the low frequencies. Conversely, if the reference beam is adjusted to match the lower intensity present in the higher frequency area of the object beam, the lower frequencies will have much less diffraction efficiency relative to the high frequencies. When the diffraction efficiency is distorted in this way, the reconstructed image will be a corrupted representation of the original object, and as a result, incorrect bit values may be retrieved from the stored data. In addition, this modulation mismatch causes the resulting hologram to have a lower overall diffraction efficiency. Given a fixed amount of laser power, such a reduction in overall diffraction efficiency decreases the attainable read-out rate of the hologram, and thus it limits the rate at which data can be transferred out of a storage device incorporating the hologram.
Various attempts have been made to alleviate this problem. These attempts have been based on the principle that what defines a pattern (for purposes of visual observation or detection by photosensors) is its corresponding pattern of luminous intensity, not its complex amplitude. What distinguishes these quantities (for simplicity of presentation, polarization is here neglected) is that the field quantity described by complex amplitude has both magnitude and phase, and is thus conveniently represented as a complex number, whereas intensity is represented by the (phaseless) real number obtained by multiplying the corresponding amplitude by its complex conjugate: I=A*.multidot.A. The properties of the Fourier transform are determined, in part, by the phases of the optical wavelets arriving at the Fourier transform plane (i.e., at the back focal plane of the transforming lens or lens system). Thus, by altering the phases of these wavelets as they emanate from the data mask, it is possible to manipulate the Fourier transform without (in principle) affecting the intensity distribution in the reconstructed image.
For example, U.S. Pat. No. 3,604,778, issued to C. B. Burckhardt on Sep. 14, 1971, describes the use of a phase mask to distribute the illumination more uniformly over the Fourier transform plane. This phase mask consists of an array of transparent elements. In use, the phase mask is juxtaposed with the data mask or, alternatively, it is projected back onto the data mask by some of the transforming lenses. One example of the latter arrangement is shown in FIG. 1. There, it is seen that lenses 15 and 20 project spatial light modulator 10 with unity (-1) magnification onto phase mask 70. It will be recalled that in the view of the figure, the phase mask lies one focal length to the left of lens 20, the spatial light modulator lies one focal length to the right of lens 15, and the separation between these lenses is equal to the sum of their respective focal lengths. The effect of this combination of lenses is to exactly image (upside down) spatial light modulator 10 onto phase mask 70.
Conventionally, the juxtaposition or projection is carried out such that each data element of the data mask lies adjacent to a corresponding element of the phase mask. Approximately one-half the elements of the phase mask, randomly selected, effect a 180.degree. (.pi.-radian) phase shift in the beam emerging from the corresponding data element. The other (approximately) one-half of the phase-mask elements do not effect a substantial phase shift.
When the phase of an array of light beams has been shifted by any phase mask, the resulting intensity distribution in the Fourier transform plane is modified by convolution of the corresponding electric field distribution with the Fourier transform of the phase mask. The effect of this in the present case is to add to the distribution broad- band noise, which randomizes phases of wavefronts that would otherwise reinforce each other through constructive interference at the dc spot.
The pattern shown in FIG. 2 is a schematic representation of the central portion of a Fourier transform. This Fourier transform results from a conventional combination of a data mask and a phase mask, each having square pixels. In the figure, brighter regions are represented by lighter shading.
Region 200 of this pattern is referred to as the central order of the Fourier transform. Relatively bright region 210, centrally located within region 200, represents the lowest spatial frequencies which, as noted, tend to contribute relatively high intensities to the pattern.
The actual size and spacing of the radial sequential orders (e.g., orders 200, 215, 220, respectively) that make up this pattern is dependent on the size and pitch (i.e., the center-to-center distance between pixels) of the pixel arrays, the amplitude and phase content of these arrays, the nature of the optical elements used, and the wavelength of the light used for illumination.
Although useful, this technique has been found disadvantageous because it results in an increase in the least acceptable size of the resulting hologram. That is, the size of the hologram is typically limited by masking the recording medium with an iris of suitable size and shape. The iris is juxtaposed with the recording medium or, alternatively, projected onto it by an imaging system that also relays the object beam onto the same medium. It is often desirable to confine the hologram in this manner, so that many such holograms can be recorded, side-by-side, or even partially overlapping, within the same medium. This makes it possible to economically store much more data than can be stored in a single hologram. The achievable amount of information that can be stored on a planar medium of a given area is inversely proportional to the square of the (linear) size of a single hologram. Thus, spreading the hologram by a factor a decreases the achievable information storage density by a factor a.sup.2.
As is well known, the image information stored in the hologram is globally distributed; that is, the complete image can be reconstructed from even a relatively small part of the hologram. However, there are both theoretical and practical limits that dictate a lower bound to the area of the hologram from which an image can be reconstructed with a desired degree of fidelity to the original object. In applications for data storage, the desired degree of fidelity is expressed by a desired maximum rate of bit error. That is, a data element having the logical value 1 should be reproduced as such in the reconstructed image, and similarly for a 0-valued element. The bit-error rate is the proportion of elements in the reconstructed image that bear the wrong logical value.
When methods are employed of the kind described in the Burckhardt patent, cited above, the smallest acceptable hologram is typically almost doubled in linear dimension relative to the case of a data mask without a phase mask.
In fact, the random binary phase mask disclosed in Burckhardt, as well as random n-valued phase masks disclosed by others (n an integer greater than 2), have been used for convenience, and not because they provide an optimal tradeoff between intensity redistribution and the size of the resulting hologram. Until now, practitioners have failed to show how such an optimal tradeoff can be achieved.